Ergodicity and conservativity of products of infinite transformations and their inverses

Julien Clancy¹ ; Rina Friedberg² ; Indraneel Kasmalkar³ ; Isaac Loh⁴ ; Tudor Pădurariu⁵ ; Cesar E. Silva⁶ ; Sahana Vasudevan⁷

¹ Yale University New Haven, CT 06520, U.S.A.
² University of Chicago Chicago, IL 60637, U.S.A.
³ University of California, Berkeley Berkeley, CA 94720, U.S.A.
⁴ Williams College Williamstown, MA 01267, U.S.A.
⁵ University of California Los Angeles, CA 90095-1555, U.S.A.
⁶ Department of Mathematics Williams College Williamstown, MA 01267, U.S.A.
⁷ Harvard University Cambridge, MA 02138, U.S.A.

Colloquium Mathematicum, Tome 143 (2016) no. 2, pp. 271-291.

Résumé

We construct a class of rank-one infinite measure-preserving transformations such that for each transformation $T$ in the class, the cartesian product $T\times T$ is ergodic, but the product $T\times T^{-1}$ is not. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.

Détail

DOI : 10.4064/cm6482-10-2015

Keywords: construct class rank one infinite measure preserving transformations each transformation class cartesian product times ergodic product times prove product rank one transformation its inverse conservative while there infinite measure preserving conservative ergodic markov shifts whose product their inverse conservative